integral of x^3e^{13x}

解答

\int x^{3} e^{13 x} d x

\frac{1}{28561} (2197 e^{13 x} x^{3} - 3 (169 e^{13 x} x^{2} - 2 (13 e^{13 x} x - e^{13 x}))) + C

求解步骤

\int x^{3} e^{13 x} d x

使用换元积分法

= \int \frac{e ^{u} u ^{3}}{28561} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{28561} \cdot \int e^{u} u^{3} d u

使用分布积分法

= \frac{1}{28561} (u^{3} e^{u} - \int 3 u^{2} e^{u} d u)

\int 3 u^{2} e^{u} d u = 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u}))

= \frac{1}{28561} (u^{3} e^{u} - 3 (u^{2} e^{u} - 2 (e^{u} u - e^{u})))

u = 13 x

代回

= \frac{1}{28561} ((13 x)^{3} e^{13 x} - 3 ((13 x)^{2} e^{13 x} - 2 (e^{13 x} \cdot 13 x - e^{13 x})))

化简

\frac{1}{28561} ((13 x)^{3} e^{13 x} - 3 ((13 x)^{2} e^{13 x} - 2 (e^{13 x} \cdot 13 x - e^{13 x}))) : \frac{1}{28561} (2197 e^{13 x} x^{3} - 3 (169 e^{13 x} x^{2} - 2 (13 e^{13 x} x - e^{13 x})))

= \frac{1}{28561} (2197 e^{13 x} x^{3} - 3 (169 e^{13 x} x^{2} - 2 (13 e^{13 x} x - e^{13 x})))

解答补常数

= \frac{1}{28561} (2197 e^{13 x} x^{3} - 3 (169 e^{13 x} x^{2} - 2 (13 e^{13 x} x - e^{13 x}))) + C

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